Magnetic filaments: shapes, stability and self-propulsion

Flexible magnetic rods are studied from different points of view. For the rod with remanent magnetization and paramagnetic susceptibility the equilibrium configurations with free and unclamped ends are calculated. It is shown by the analysis of the corresponding eigenvalue problems that these shapes are unstable. In order to avoid this instability when the ferromagnetic swimmer is realized experimentally and exhibits the self-propelling motion that occurs due to the U-like deformation and its relaxation, the duration of the pulse of the driving magnetic field during which the deformation develops is shortened relative to the part of the cycle that corresponds to the relaxation of the U-like shape. The experimentally observed dynamics of the shapes is in good agreement with numerical simulations if the ratio of the perpendicular and parallel drag coefficients is taken as 1.3 instead of 2. The mechanism of propulsion of the ferromagnetic rod is similar to that of Chlamydomonas and is based on the non-reciprocity of the bending and relaxation stages as found both experimentally and numerically.